Class 07

Last time…

• The normal distribution and when to use the mean + standard deviation

• Robust statistics and when to use the median + interquartile range

• The 5-number summary and how to read a boxplot

• Other distribution plots: dot plot, histogram, density plot, violin plot

Practice: Visualizing Distributions

Practice: Visualizing Distributions

Practice: Summary Statistics

Is the mean greater than, lesser than, or about equal to the mean? What does that mean for the shape of the distribution?

Practice: Summary Statistics

Today

• Analyze contingency (e.g. 2x2) tables

• Summarizing categorical variables with proportions

• Visualizing data using categories

• Comparing data from 2+ variables

What is a contingency table?

A contingency table is a cross-tabulation of the frequency of observations across 2 categorical variables

In this table, the values are the count of observations that belong both the corresponding row and column categories.

What is frequency?

When someone describes a category’s frequency, they may be referring to…

• Count: the total number of times the category appears in the data

• Proportion: the total number of times the category appears in the data divided by the number of observations

Calculating a proportion

Only a subset of observations belong to each level of a categorical variable. A proportion describes the count of the observations belonging to that category divided by the total number of observations (\(n\)).

\[ \operatorname{Proportion} = \frac{\operatorname{count}(\operatorname{category})}{n} \]

Proportions can be any real number between a minimum of 0 (\(\frac{0}{n}\)) and maximum of 1 (\(\frac{n}{n}\)).

Calculating Proportions by row

The row totals are all 1, which is the maximum value of a proportion. This indicates that the denominator for the proportions is the row total for each cell.

Calculating Proportions by Column

The column totals are all 1, which is the maximum value of a proportion. This indicates that the denominator for the proportions is the column total for each cell.

Example: Field Goals

In the NFL, kickers can play a pivotal role in determining the outcome of a game. Much time and money has been devoted to determining the factors which lead to a successful kick. Past performance is one factor which could affect whether a kick is good or not. Are kickers more or less likely to make the extra point kick if they missed their last kick?

Contingency Table: Counts

Contingency Table: Proportions by Row

Contingency Table: Proportions by Column

Visualizing Frequencies: Bar Plot (Counts)

Visualizing Frequencies: Bar Plot (Proportions)

Comparing 2 (Categorical) Variables: Side-By-Side Bar Plot

Comparing 2 (Categorical) Variables: Stacked Bar Plot (counts)

Comparing 2 (Categorical) Variables: Stacked Bar Plot (proportions)

Bar plots in the news this week…

Comparing 2 (Categorical) Variables: Mosaic Plot (2 Variables)

Comparing a categorical variable and a numeric variable

• Histograms + density curves

• Boxplot + violin plot

• Rain cloud plot

Which type of plot(s) are shown in these figures?

Which type of plot(s) are shown in these figures?

Distributions in the news this week…

Comparing 2 Numerical Variables: Scatter plot

More ways to describe the relationship between 2 numerical variables

• Is the relationship between the 2 variables linear or nonlinear ?

• Is the relationship strong or weak?

Example: describing numerical variable relationships

Visualizing the relationship between 2 numerical variables: add a regression line!

Does this line better “fit” the data?

Linear vs Nonlinear Relationships

Can I compare 3 or more variables in a single plot?

Changing shapes and sizes of points

Split into separate plots with faceting

Homework 2

• Describing numerical distributions: modality, skew, outliers

• Describing numerical distributions: appropriate summary statistics

• Matching numerical distributions to their summary statistics, reading a boxplot

• Calculating proportions from a contingency table

Next Week: Chapter 3, Probability

• What is probability?

• Disjoint vs not disjoint sets

• Probability distributions

• Complements and independence